arXiv research

This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the persistence of the features of changing shapes. The basic approach …

This article introduces planar shape signatures derived from homology nerves, which are intersecting 1-cycles in a collection of homology groups endowed with a proximal relator (set of nearness relations) that includes a descriptive proximity. A 1-cycle is a closed, connected path with a zero boundary in a simplicial c…

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk …

The boundary of certain hyperbolic groups is like a Menger curve.

We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.

The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.

This article introduces vortex nerve complexes in CW (Closure finite Weak) topological spaces, which first appeared in works by P. Alexandroff, H. Hopf and J.H.C. Whitehead during the 1930s. A vortex nerve is a CW complex containing one or more intersecting path-connected cycles. Each vortex nerve has its own distincti…

Graev's nerve implies invariant Einstein metrics on homogeneous spaces.

New right-angled Artin subgroups found in Artin groups.

Polyhedral semantics for intermediate logics; Nerve Criterion ensures completeness.

Using ideas of the Dowker duality we prove that the Rips complex at scale $r$ is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. Thes…

Characterizes Coxeter groups with specific boundary shapes.

Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös-Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about t…

The paper bridges diffeological bundle theory with higher topos theory.

We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series…

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| …

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not …

Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $\ell^2$-homology of Sigma vanishes in all but the middle dimension.

We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these …

We give a proof of the Singer conjecture (on the vanishing of reduced $\ell^2$-homology except in the middle dimension) for the Davis Complex $Σ$ associated to a Coxeter system $(W,S)$ whose nerve $L$ is a triangulation of $\mathbb{S}^2$. We show that it follows from a theorem of Andreev, which gives the necessary and …

This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's …

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve …

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its si…

Mapping class group subgroups yield quasi-isometric curve complex.

For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpiński compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the …

Unified framework for Morita invariant cohomology of Lie groupoids.

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending …

The \emph{action dimension} of a discrete group $G$ is the minimum dimension of a contractible manifold, which admits a proper $G$-action. In this paper, we study the action dimension of general Artin groups. The main result is that the action dimension of an Artin group with the nerve $L$ of dimension $n$ for $n \ne 2…

The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $\mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $\mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded into $(2n+1)$-dimensional …

Constructs hyperbolic reflection groups with 3D limit sets.

New results on relative simplicial volume using bounded acyclicity.

We give a complete computation of the BNSR-invariants $Σ^m(H_n)$ of the Houghton groups $H_n$. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated $CAT(0)$ cube complex by their interse…

Given a Coxeter system $(W,S)$ and a multiparameter $\mathbf{q}$ of real numbers indexed by $S$, one can define the weighted $L^2$-cohomology groups and associate to them a nonnegative real number called the weighted $L^2$-Betti number. We show that for ranges of $\mathbf{q}$ depending on certain subgroups of $W$, the …

Suppose that $(W,S)$ is a Coxeter system with associated Artin group $A$ and with a simplicial complex $L$ as its nerve. We define the notion of a "standard abelian subgroup" in $A$. The poset of such subgroups in $A$ is parameterized by the poset of simplices in a certain subdivision $L_\oslash$ of $L$. This complex o…

To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g-2)-spheres. This…

We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms introduced by V.Hinich.

We generalize the methods in previous work to provide a program for proving Singer's Conjecture for Coxeter systems. Specifically, we consider even Coxeter systems with nerves that are flag triangulations of $\BS^{n-1}$, $n=2k$. We prove that Singer's Conjecture in dimensions $n-2$ and $n-1$, along with the vanishing o…

Glaucoma is a serious ocular disorder for which the screening and diagnosis are carried out by the examination of the optic nerve head (ONH). The color fundus image (CFI) is the most common modality used for ocular screening. In CFI, the central r

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of usi…

The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infin…

Overlays were introduced by R. H. Fox [6] as a subclass of covering maps. We offer a different view of overlays: it resembles the definition of paracompact spaces via star refinements of open covers. One introduces covering structures for covering maps and $p:X\to Y$ is an overlay if it has a covering structure that ha…

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spac…

Glaucoma is the second leading cause of blindness all over the world, with approximately 60 million cases reported worldwide in 2010. If undiagnosed in time, glaucoma causes irreversible damage to the optic nerve leading to blindness. The optic nerve head examination, which involves measurement of cup-to-disc ratio, is…

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a c…

We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…

A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are …

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated l…

We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories, showing that it yields simplicial manifolds if the 2-cells are invertible. Finally, o…